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 A086990 Expansion of (1+4x-sqrt(1+4x^2))/(4+6x) in powers of x. 3
 0, 1, -2, 3, -4, 6, -10, 15, -20, 30, -52, 78, -96, 144, -282, 423, -420, 630, -1660, 2490, -1304, 1956, -11332, 16998, 3896, -5844, -95240, 142860, 157160, -235740, -983610, 1475415, 2634300, -3951450, -11751660, 17627490, 38381160, -57571740 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Series reversion of Sum_{k>=0} a(k)x^k is x(Sum_{k>=0} A007051(k)x^k). G.f. A(x) = Sum_{k>=0} a(k)x^k satisfies 0 = x - (4x+1)*A(x) + (3x+2)*A(x)^2. If A(x)=g.f., then y=x/A(x)-2x satisfies x^2 = y^2 - y. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 FORMULA G.f.: (1+4x-sqrt(1+4x^2))/(4+6x). G.f.: (x-x^2*c(-x^2))/(1+x-x^2*c(-x^2)), c(x) the g.f. of A000108. - Paul Barry, Jun 17 2005 From Gary W. Adamson, Jan 05 2012: (Start) a(n) is the upper left term of (-1)*M^n, where M = an infinite square production matrix as follows: -1, -1, 0, 0, 0, 0, ... -1, 1, -1, 0, 0, 0, ... -1, 1, 1, -1, 0, 0, ... -1, 1, 1, 1, -1, 0, ... -1, 1, 1, 1, 1, -1, ... ... (End) D-finite with recurrence 2*n*a(n) +3*n*a(n-1) +8*(n-3)*a(n-2) +12*(n-3)*a(n-3)=0. - R. J. Mathar, Nov 24 2012 Lim sup n->infinity |a(n)|^(1/n) = 2. - Vaclav Kotesovec, Feb 09 2014 EXAMPLE a(5) = 6 = upper left term of (-1)*M^5. - Gary W. Adamson, Jan 05 2012 MATHEMATICA CoefficientList[Series[(1 + 4 x - Sqrt[1 + 4 x^2])/(4 + 6 x), {x, 0, 50}], x] (* Harvey P. Dale, Mar 24 2011 *) PROG (PARI) a(n)=polcoeff((1+4*x-sqrt(1+4*x^2+x*O(x^n)))/(4+6*x), n) CROSSREFS Sequence in context: A173473 A288807 A097699 * A090412 A073028 A147788 Adjacent sequences: A086987 A086988 A086989 * A086991 A086992 A086993 KEYWORD sign AUTHOR Michael Somos, Jul 27 2003 STATUS approved

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Last modified December 3 21:20 EST 2023. Contains 367540 sequences. (Running on oeis4.)